1. INTRODUCTION TO LOGIC - NOTES

1.1: BASIC CONCEPTS

Logic may be defined as the science that evaluates arguments. All of us encounter arguments in our day-

to-day experience. We read them in books and newspapers, hear them on television, and formulate them

when communicating with friends and associates. The aim of logic is to develop a system of methods and

principles that we may use as criteria for evaluating the arguments of others and as guides in constructing

arguments of our own. Its purpose, as the science that evaluates arguments, is thus to develop methods

and techniques that allow us to distinguish good arguments from bad. Among the benefits to be expected

from the study of logic is an increase in confidence that we are making sense when we criticize the

arguments of others and when we advance arguments of our own.

An argument, as it occurs in logic, is a group of statements, one or more of which (the premises) are

claimed to provide support for, or reasons to believe, one of the others (the conclusion). All arguments

may be placed in one of two basic groups: those in which the premises really do support the conclusion

and those in which they do not, even though they are claimed to. The former are said to be good

arguments (at least to that extent), the latter bad arguments.

As is apparent from the above definition, the term ‘‘argument’’ has a very specific meaning in logic. It

does not mean, for example, a mere verbal fight, as one might have with one’s parent, spouse, or friend.

Let us examine the features of this definition in greater detail. First of all, an argument is a group of

statements.

A statement is a sentence that is either true or false—in other words, typically a declarative sentence or a

sentence component that could stand as a declarative sentence. The following sentences are statements:

Aluminum is attacked by hydrochloric acid.

Broccoli is a good source of vitamin A.

Argentina is located in North America.

Napoleon prevailed at Waterloo.

Rembrandt was a painter and Shelley was a poet.

The first two statements are true, the second two false. The last one expresses two statements, both of

which are true. Truth and falsity are called the two possible truth values of a statement. Thus, the truth

value of the first two statements is true, the truth value of the second two is false, and the truth value of

the last statement, as well as that of its components, is true.

Unlike statements, many sentences cannot be said to be either true or false. Questions, proposals,

suggestions, commands, and exclamations usually cannot, and so are not usually classified as statements.

The following sentences are not statements:

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What is the atomic weight of carbon? (question)

Let’s go to the park today (proposal)

We suggest that you travel by bus (suggestion)

Turn to the left at the next corner (command)

All right! (exclamation)

The statements that make up an argument are divided into one or more premises and one and only one

conclusion. The premises are the statements that set forth the reasons or evidence, and the conclusion is

the statement that the evidence is claimed to support or imply. In other words, the conclusion is the

statement that is claimed to follow from the premises. Here is an example of an argument:

All crimes are violations of the law/Theft is a crime//Therefore, theft is a violation of the law.

The first two statements are the premises; the third is the conclusion. (The claim that the premises support

or imply the conclusion is indicated by the word ‘‘therefore.’’) In this argument the premises really do

support the conclusion, and so the argument is a good one. But consider this argument:

Some crimes are misdemeanors/Murder is a crime//Therefore, murder is a misdemeanor.

In this argument the premises do not support the conclusion, even though they are claimed to, and so the

argument is not a good one.

One of the most important tasks in the analysis of arguments is being able to distinguish premises from

conclusion. If what is thought to be a conclusion is really a premise, and vice versa, the subsequent

analysis cannot possibly be correct. Frequently, arguments contain certain indicator words that provide

clues in identifying premises and conclusion. Some typical conclusion indicators are:

Therefore, wherefore, accordingly, we may conclude, entails that, hence, thus, consequently, we

may infer, it must be that, whence, so, it follows that, implies that, as a result

Whenever a statement follows one of these indicators, it can usually be identified as the conclusion. By

process of elimination the other statements in the argument are the premises.

Example: Corporate raiders leave their target corporation with a heavy debt burden and no increase in productive capacity.

Consequently, corporate raiders are bad for the business community.

The conclusion of this argument is ‘‘Corporate raiders are bad for the business community,’’ and the

premise is ‘‘Corporate raiders leave their target corporation with a heavy debt burden and no increase in

productive capacity.’’

If an argument does not contain a conclusion indicator, it may contain a premise indicator. Some typical

premise indicators are:

Since, as indicated by, because, for, in that, may be inferred from, as, given that, seeing that, for the

reason that, inasmuch as, owing to

Any statement following one of these indicators can usually be identified as a premise.

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Example: Expectant mothers should never use recreational drugs, since the use of these drugs can jeopardize the development

of the fetus.

The premise of this argument is ‘‘the use of these drugs can jeopardize the development of the fetus,’’

and the conclusion is ‘‘Expectant mothers should never use recreational drugs.’’

One premise indicator not included in the above list is ‘‘for this reason.’’ This indicator is special in that

it comes immediately after the premise that it indicates. ‘‘For this reason’’ (except when followed by a

colon) means for the reason (premise) that was just given. In other words, the premise is the statement

that occurs immediately before ‘‘for this reason.’’ One should be careful not to confuse ‘‘for this reason’’

with ‘‘for the reason that.’’

Sometimes a single indicator can be used to identify more than one premise. Consider the following

argument:

The development of high-temperature superconducting materials is technologically justified, for such materials will allow

electricity to be transmitted without loss over great distances, and they will pave the way for trains that levitate magnetically.

The premise indicator ‘‘for’’ goes with both ‘‘such materials will allow electricity to be transmitted

without loss over great distances’’ and ‘‘they will pave the way for trains that levitate magnetically.’’

These are the premises. By process of elimination, ‘‘the development of high-temperature

superconducting materials is technologically justified’’ is the conclusion.

Sometimes an argument contains no indicators. When this occurs, the reader/listener must ask himself or

herself such questions as: What single statement is claimed (implicitly) to follow from the others? What is

the arguer trying to prove? What is the main point in the passage? The answers to these questions should

point to the conclusion.

Example: The space program deserves increased expenditures in the years ahead. Not only does the national defense depend

upon it, but the program will more than pay for itself in terms of technological spinoffs. Furthermore, at current funding levels

the program cannot fulfill its anticipated potential.

The conclusion of this argument is the first statement, and all of the other statements are premises. The

argument illustrates the pattern found in most arguments that lack indicator words: the intended

conclusion is stated first, and the remaining statements are then offered in support of this first statement.

When the argument is restructured according to logical principles, however, the conclusion is always

listed after the premises:

P1: The national defense is dependent upon the space program.

P2: The space program will more than pay for itself in terms of technological spinoffs.

P3: At current funding levels the space program cannot fulfill its anticipated potential.

C: The space program deserves increased expenditures in the years ahead.

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When restructuring arguments such as this, one should remain as close as possible to the original version,

while at the same time attending to the requirement that premises and conclusion be complete sentences

that are meaningful in the order in which they are listed.

Note that the first two premises are included within the scope of a single sentence in the original

argument. For the purposes of this chapter, compound arrangements of statements in which the various

components are all claimed to be true will be considered as separate statements.

Passages that contain arguments sometimes contain statements that are neither premises nor conclusion.

Only statements that are actually intended to support the conclusion should be included in the list of

premises. If a statement has nothing to do with the conclusion or, for example, simply makes a passing

comment, it should not be included within the context of the argument.

Example: Socialized medicine is not recommended because it would result in a reduction in the overall quality of medical care

available to the average citizen. In addition, it might very well bankrupt the federal treasury. This is the whole case against

socialized medicine in a nutshell.

The conclusion of this argument is ‘‘Socialized medicine is not recommended,’’ and the two statements

following the word ‘‘because’’ are the premises. The last statement makes only a passing comment about

the argument itself and is therefore neither a premise nor a conclusion.

Closely related to the concepts of argument and statement are those of inference and proposition. An

inference, in the technical sense of the term, is the reasoning process expressed by an argument. As we

will see in the next section, inferences may be expressed not only through arguments but through

conditional statements as well.

In the loose sense of the term, ‘‘inference’’ is used interchangeably with ‘‘argument.’’ Analogously, a

proposition, in the technical sense, is the meaning or information content of a statement. For the purposes

of this book, however, ‘‘proposition’’ and ‘‘statement’’ are used interchangeably.

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Note on the History of Logic

The person who is generally credited as being the father of logic is the ancient Greek philosopher

Aristotle (384–322 B.C.). Aristotle’s predecessors had been interested in the art of constructing

persuasive arguments and in techniques for refuting the arguments of others, but it was Aristotle who first

devised systematic criteria for analyzing and evaluating arguments. Aristotle’s logic is called syllogistic

logic and includes much of what is treated in Chapters 4 and 5 of this text. The fundamental elements in

this logic are terms, and arguments are evaluated as good or bad depending on how the terms are arranged

in the argument. In addition to his development of syllogistic logic, Aristotle cataloged a number of

informal fallacies, a topic treated in Chapter 3 of this text.

After Aristotle’s death, another Greek philosopher, Chrysippus (279–206 B.C.), one of the founders of the

Stoic school, developed a logic in which the fundamental elements were whole propositions. Chrysippus

treated every proposition as either true or false and developed rules for determining the truth or falsity of

compound propositions from the truth or falsity of their components. In the course of doing so, he laid the

foundation for the truth functional interpretation of the logical connectives presented in Chapter 6 of this

text and introduced the notion of natural deduction, treated in Chapter 7.

For thirteen hundred years after the death of Chrysippus, relatively little creative work was done in logic.

The physician Galen (A.D. 129–ca. 199) developed the theory of the compound categorical syllogism, but

for the most part philosophers confined themselves to writing commentaries on the works of Aristotle and

Chrysippus. Boethius (ca. 480–524) is a noteworthy example.

The first major logician of the Middle Ages was Peter Abelard (1079–1142). Abelard reconstructed and

refined the logic of Aristotle and Chrysippus as communicated by Boethius, and he originated a theory of

universals that traced the universal character of general terms to concepts in the mind rather than to

‘‘natures’’ existing outside the mind, as Aristotle had held. In addition, Abelard distinguished arguments

that are valid because of their form from those that are valid because of their content, but he held that only

formal validity is the ‘‘perfect’’ or conclusive variety. The present text follows Abelard on this point.

After Abelard, the study of logic during the Middle Ages blossomed and flourished through the work of

numerous philosophers. It attained its final expression in the writings of the Oxford philosopher William

of Occam (ca. 1285–1349). Occam devoted much of his attention to modal logic, a kind of logic that

involves such notions as possibility, necessity, belief, and doubt. He also conducted an exhaustive study

of forms of valid and invalid syllogisms and contributed to the development of the concept of a

metalanguage—that is, a higher-level language used to discuss linguistic entities such as words, terms,

propositions, and so on.

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Toward the middle of the fifteenth century, a reaction set in against the logic of the Middle Ages.

Rhetoric largely displaced logic as the primary focus of attention; the logic of Chrysippus, which had

already begun to lose its unique identity in the Middle Ages, was ignored altogether, and the logic of

Aristotle was studied only in highly simplistic presentations. A reawakening did not occur until two

hundred years later through the work of GottfriedWilhelm Leibniz (1646–1716).

Leibniz, a genius in numerous fields, attempted to develop a symbolic language or ‘‘calculus’’ that could

be used to settle all forms of disputes, whether in theology, philosophy, or international relations. As a

result of this work, Leibniz is sometimes credited with being the father of symbolic logic. Leibniz’s

efforts to symbolize logic were carried into the nineteenth century by Bernard Bolzano (1781–1848).

With the arrival of the middle of the nineteenth century, logic commenced an extremely rapid period of

development that has continued to this day. Work in symbolic logic was done by a number of

philosophers and mathematicians, includingAugustus DeMorgan (1806–1871), George Boole (1815–

1864), William Stanley Jevons (1835–1882), and John Venn (1834–1923), some of whom are popularly

known today by the logical theorems and techniques that bear their names. At the same time, a revival in

inductive logic was initiated by the British philosopher John Stuart Mill (1806–1873), whose methods of

induction are presented in Chapter 9 of this text.

Toward the end of the nineteenth century, the foundations of modern mathematical logic were laid by

Gottlob Frege (1848–1925). His Begriffsschrift sets forth the theory of quantification presented in Chapter

8 of this text. Frege’s work was continued into the twentieth century by Alfred North Whitehead (1861–

1947) and Bertrand Russell (1872–1970), whose monumental Principia Mathematica attempted to reduce

the whole of pure mathematics to logic. The Principia is the source of much of the symbolism that

appears in Chapters 6, 7, and 8 of this text.

During the twentieth century, much of the work in logic has focused on the formalization of logical

systems and on questions dealing with the completeness and consistency of such systems. A now-famous

theorem proved by Kurt Goedel (1906–1978) states that in any formal system adequate for number theory

there exists an undecidable formula—that is, a formula such that neither it nor its negation is derivable

from the axioms of the system. Other developments include multivalued logics and the formalization of

modal logic. Most recently, logic has made a major contribution to technology by providing the

conceptual foundation for the electronic circuitry of digital computers.

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1.2: Recognizing Arguments

Not all passages contain arguments. Because logic deals with arguments, it is important to be able to

distinguish passages that contain arguments from those that do not.

In general, a passage contains an argument if it purports to prove something; if it does not do so, it does

not contain an argument. Two conditions must be fulfilled for a passage to purport to prove something:

(1) At least one of the statements must claim to present evidence or reasons. (2) There must be a claim

that the alleged evidence or reasons supports or implies something—that is, a claim that something

follows from the alleged evidence. As we have seen the statements that claim to present the evidence or

reasons are the premises and the statement that the evidence is claimed to support or imply is the

conclusion. It is not necessary that the premises present actual evidence or true reasons nor that the

premises actually support the conclusion. But at least the premises must claim to present evidence or

reasons, and there must be a claim that the evidence or reasons support or imply something.

The first condition expresses a factual claim, and deciding whether it is fulfilled usually presents few

problems. Thus, most of our attention will be concentrated on whether the second condition is fulfilled.

This second condition expresses what is called an inferential claim. The inferential claim is simply the

claim that the passage expresses a certain kind of reasoning process—that something supports or implies

something or that something follows from something. Such a claim can be either explicit or implicit.

An explicit inferential claim is usually asserted by premise or conclusion indicator words (‘‘thus,’’

‘‘since,’’ ‘‘because,’’ ‘‘hence,’’ ‘‘therefore,’’ and so on).

Example: The human eye can see a source of light that is as faint as an ordinary candle from a distance of 27 kilometers,

through a non-absorbing atmosphere. Thus, a powerful searchlight directed from a new moon should be visible on earth with the

naked eye.(Diane E. Papalia and Sally Wendkos Olds, Psychology)

The word ‘‘thus’’ expresses the claim that something is being inferred, so the passage is an argument.

An implicit inferential claim exists if there is an inferential relationship between the statements in a

passage.

Example: The price reduction [seen with the electronic calculator] is the result of a technological revolution. The calculator of

the 1960s used integrated electronic circuits that contained about a dozen transistors or similar components on a single chip.

Today, mass-produced chips, only a few millimeters square, contain several thousand such components. (Robert S. Boikess and

Edward Edelson, Chemical Principles)

The inferential relationship between the first statement and the other two constitutes an implicit claim that

evidence supports something, so we are justified in calling the passage an argument. The first statement is

the conclusion, and the other two are the premises.

In deciding whether there is a claim that evidence supports or implies something, keep an eye out for: (1)

indicator words and (2) the presence of an inferential relationship between the statements. In connection

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with these points, however, a word of caution is in order. First, the mere occurrence of an indicator word

by no means guarantees the presence of an argument.

For example, consider the following passages:

Since Edison invented the phonograph; there have been many technological developments. Since Edison invented the

phonograph, he deserves credit for a major technological development.

In the first passage the word ‘‘since’’ is used in a temporal sense. It means ‘‘from the time that.’’ Thus,

the first passage is not an argument. In the second passage ‘‘since’’ is used in a logical sense, and so the

passage is an argument.

The second cautionary point is that it is not always easy to detect the occurrence of an inferential

relationship between the statements in a passage, and the reader may have to review a passage several

times before making a decision. In reaching such a decision, it sometimes helps to mentally insert the

word ‘‘therefore’’ before the various statements to see whether it makes sense to interpret one of them as

following from the others. Even with this mental aid, however, the decision whether a passage contains an

inferential relationship (as well as the decision about indicator words) often involves a heavy dose of

interpretation. As a result, not everyone will agree about every passage. Sometimes the only answer

possible is a conditional one: ‘‘If this passage contains an argument, then these are the premises and that

is the conclusion.’’ To assist in distinguishing passages that contains arguments from those that do not, let

us now investigate some typical kinds of non-arguments. These include simple non-inferential passages,

expository passages, illustrations, explanations, and conditional statements.

Simple Non-inferential Passages

Simple noninferential passages are unproblematic passages that lack a claim that anything is being

proved. Such passages contain statements that could be premises or conclusions (or both), but what is

missing is a claim that any potential premise supports a conclusion or that any potential conclusion is

supported by premises. Passages of this sort include warnings, pieces of advice, statements of belief or

opinion, loosely associated statements, and reports.

A warning is a form of expression that is intended to put someone on guard against a dangerous or

detrimental situation.

Examples: Watch out that you don’t slip on the ice. Whatever you do, never confide personal secrets to Blabbermouth Bob.

If no evidence is given to prove that such statements are true, then there is no argument.

A piece of advice is a form of expression that makes a recommendation about some future decision or

course of conduct.

Examples: You should keep a few things in mind before buying a used car. Test drive the car at varying speeds and

conditions, examine the oil in the crankcase, ask to see service records, and, if possible, have the engine and power train checked

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by a mechanic. Before accepting a job after class hours, I would suggest that you give careful consideration to your course load.

Will you have sufficient time to prepare for classes and tests, and will the job produce an excessive drain on your energies?

As with warnings, if there is no evidence that is intended to prove anything, then there is no argument.

A statement of belief or opinion is an expression about what someone happens to believe or think at a

certain time.

Examples: We believe that our company must develop and produce outstanding products that will perform a great service or

fulfill a need for our customers.We believe that our business must be run at an adequate profit and that the services and products

we offer must be better than those offered by competitors. (Robert D. Hay and Edmund R. Gray, ‘‘Introduction to Social

Responsibility’’)

I think a nation such as ours, with its high moral traditions and commitments, has a further responsibility to know how we

became drawn into this conflict, and to learn the lessons it has to teach us for the future. (Alfred Hassler, Saigon, U.S.A.)

Because neither of these authors makes any claim that his belief or opinion is supported by evidence, or

that it supports some conclusion, there is no argument.

Loosely associated statements may be about the same general subject, but they lack a claim that one of

them is proved by the others.

Example: Not to honor men of worth will keep the people from contention; not to value goods that are hard to come by will

keep them from theft; not to display what is desirable will keep them from being unsettled of mind. (Lao-Tzu, Thoughts from the

Tao Te Ching)

Because there is no claim that any of these statements provides evidence or reasons for believing another,

there is no argument.

A report consists of a group of statements that convey information about some topic or event.

Example: Even though more of the world is immunized than ever before, many old diseases have proven quite resilient in the

face of changing population and environmental conditions, especially in the developing world. New diseases, such as AIDS, have

taken their toll in both the North and the South. (Steven L. Spiegel, World Politics in a New Era)

These statements could serve as the premises of an argument; but because the author makes no claim that

they support or imply anything, there is no argument. Another type of report is the news report:

A powerful car bomb blew up outside the regional telephone company headquarters in Medellin, injuring 25 people and causing

millions of dollars of damage to nearby buildings, police said. A police statement said the 198-pound bomb was packed into a

milk churn hidden in the back of a stolen car. (Newspaper clipping)

Again, because the reporter makes no claim that these statements imply anything, there is no argument.

One must be careful, though, with reports about arguments:

‘‘The Air Force faces a serious shortage of experienced pilots in the years ahead, because repeated overseas tours and the allure

of high paying jobs with commercial airlines are winning out over lucrative bonuses to stay in the service,’’ says a prominent Air

Force official. (Newspaper clipping)

Properly speaking, this passage is not an argument, because the author of the passage does not claim that

anything is supported by evidence. Rather, the author reports the claim by the Air Force official that

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something is supported by evidence. If such passages are interpreted as ‘‘containing’’ arguments, it must

be made clear that the argument is not the author’s but one made by someone about whom the author is

reporting.

Expository Passages: an expository passage is a kind of discourse that begins with a topic sentence

followed by one or more sentences that develop the topic sentence. If the objective is not to prove the

topic sentence but only to expand it or elaborate it, then there is no argument.

Examples: There are three familiar states of matter: solid, liquid, and gas. Solid objects ordinarily maintain their shape and

volume regardless of their location. A liquid occupies a definite volume, but assumes the shape of the occupied portion of its

container. A gas maintains neither shape nor volume. It expands to fill completely whatever container it is in. (John W. Hill and

Doris K. Kolb, Chemistry for Changing Times, 7th ed.)

There is a stylized relation of artist to mass audience in the sports, especially in baseball. Each player develops a style of his own

—the swagger as he steps to the plate, the unique windup a pitcher has, the clean-swinging and hard-driving hits, the precision

quickness and grace of infield and outfield, the sense of surplus power behind whatever is done. (Max Lerner, America as a

Civilization)

In each passage the topic sentence is stated first, and the remaining sentences merely develop and flesh

out this topic sentence. These passages are not arguments because they lack an inferential claim.

However, expository passages differ from simple non-inferential passages (such as warnings and pieces

of advice) in that many of them can also be taken as arguments. If the purpose of the subsequent

sentences in the passage is not only to flesh out the topic sentence but also to prove it, then the passage is

an argument.

Example: Skin and the mucous membrane lining the respiratory and digestive tracts serve as mechanical barriers to entry by

microbes. Oil gland secretions contain chemicals that weaken or kill bacteria on skin. The respiratory tract is lined by cells that

sweep mucus and trapped particles up into the throat, where they can be swallowed. The stomach has an acidic pH, which

inhibits the growth of many types of bacteria. (Sylvia S. Mader. Human Biology, 4th ed.)

In this passage the topic sentence is stated first, and the purpose of the remaining sentences is not only to

show how the skin and mucous membranes serve as barriers to microbes but to prove that they do this.

Thus, the passage can be taken as both an expository passage and an argument.

In deciding whether an expository passage should be interpreted as an argument, try to determine whether

the purpose of the subsequent sentences in the passage is merely to develop the topic sentence or also to

prove it. In borderline cases, ask yourself whether the topic sentence makes a claim that everyone accepts

or agrees with. If it does, the passage is probably not an argument. In real life situations authors rarely try

to prove something that everyone already accepts. However, if the topic sentence makes a claim that

many people do not accept or have never thought about, then the purpose of the remaining sentences may

be both to prove the topic sentence as well as to develop it. If this be so, the passage is an argument.

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Finally, if even this procedure yields no definite answer, the only alternative may be to say that if the

passage is taken as an argument, then the first statement is the conclusion and the others are the premises.

Illustrations: An illustration consists of a statement about a certain subject combined with a reference to

one or more specific instances intended to exemplify that statement. Illustrations are often confused with

arguments because many of them contain indicator words such as ‘‘thus.’’

Examples: Chemical elements, as well as compounds, can be represented by molecular formulas. Thus, oxygen is represented

by ‘‘O2,’’ water by ‘‘H2O,’’ and sodium chloride by ‘‘NaCl.’’

Whenever a force is exerted on an object, the shape of the object can change. For example, when you squeeze a rubber ball or

strike a punching bag with your fist, the objects are deformed to some extent. (Raymond A. Serway, Physics For Scientists and

Engineers, 4th ed.)

These selections are not arguments because they make no claim that anything is being proved. In the first

selection, the word ‘‘thus’’ indicates how something is done—namely, how chemical elements and

compounds can be represented by formulas. In the second selection, the example cited is intended to give

concrete meaning to the notion of a force changing the shape of something. It is not intended primarily to

prove that a force can change the shape of something.

However, as with expository passages, many illustrations can be taken as arguments.

Such arguments are often called arguments from example. Here is an instance of one:

Water is an excellent solvent. It can dissolve a wide range of materials that will not dissolve in other liquids. For example, salts

do not dissolve in most common solvents, such as gasoline, kerosene, turpentine and cleaning fluids. But many salts dissolve

readily in water. So do a variety of nonionic organic substances, such as sugars and alcohols of low molecular weight. (Robert S.

Boikess and Edward Edelson, Chemical Principles)

In this passage the examples that are cited can be interpreted as providing evidence that water can

dissolve a wide range of materials that will not dissolve in other liquids.

Thus, the passage can be taken as both an illustration and an argument, with the second sentence being

the conclusion.

In deciding whether an illustration should be interpreted as an argument one must determine whether the

passage merely shows how something is done or what something means, or whether it also purports to

prove something. In borderline cases it helps to note whether the claim being illustrated is one that

practically everyone accepts or agrees with. If it is, the passage is probably not an argument. As we have

already noted, in real life situations authors rarely attempt to prove what everyone already accepts. But if

the claim being illustrated is one that many people do not accept or have never thought about, then the

passage may be interpreted as both an illustration and an argument.

Thus, in reference to the first two examples we considered, most people are aware that elements and

compounds can be expressed by formulas—practically everyone knows that water is H2O—and most

people know that forces distort things—that running into a tree can cause a dent in the car bumper. But

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people may not be aware of the fact that water dissolves many things that other solvents will not dissolve.

This is one of the reasons for evaluating the first two examples as mere illustrations and the last one as an

argument.

8. Explanations: One of the most important kinds of non-argument is the explanation. An explanation is

a group of statements that purports to shed light on some event or phenomenon. The event or

phenomenon in question is usually accepted as a matter of fact.

Examples: The Challenger spacecraft exploded after liftoff because an O-ring failed in one of the booster rockets. The sky

appears blue from the earth’s surface because light rays from the sun are scattered by particles in the atmosphere.

Cows can digest grass, while humans cannot, because their digestive systems contain enzymes not found in humans.

Every explanation is composed of two distinct components: the explanandum and explanans. The

explanandum is the statement that describes the event or phenomenon to be explained, and the

explanans is the statement or group of statements that purports to do the explaining. In the first example

above, the explanandum is the statement ‘‘The Challenger spacecraft exploded after liftoff,’’ and the

explanans is ‘‘An O-ring failed in one of the booster rockets.’’

Explanations are sometimes mistaken for arguments because they often contain the indicator word

‘‘because.’’ Yet explanations are not arguments because in an explanation the purpose of the explanans is

to shed light on, or to make sense of, the explanandum event—not to prove that it occurred. In other

words, the purpose of the explanans is to show why something is the case, while in an argument, the

purpose of the premises is to prove that something is the case.

In the first example above, the fact that the Challenger exploded is known to everyone. The statement that

an O-ring failed in one of the booster rockets is not intended to prove that the spacecraft exploded but

rather to show why it exploded. In the second example, the fact that the sky is blue is readily apparent.

The intention of the passage is to explain why it appears blue—not to prove that it appears blue.

Similarly, in the third example, virtually everyone knows that people cannot digest grass. The intention of

the passage is to explain why this is true.

Thus, to distinguish explanations from arguments, identify the statement that is either the explanandum or

the conclusion (usually this is the statement that precedes the word ‘‘because’’). If this statement

describes an accepted matter of fact, and if the remaining statements purport to shed light on this

statement, then the passage is an explanation.

This method works for practically all passages that are either explanations or arguments (but not both).

However, as with expository passages and illustrations, there are some passages that can be interpreted as

both explanations and arguments.

Example: Women become intoxicated by drinking a smaller amount of alcohol than men because men

metabolize part of the alcohol before it reaches the bloodstream whereas women do not.

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The purpose of this passage could be to prove the first statement to those people who do not accept it as

fact, and to shed light on that fact to those people who do accept it. Alternately, the passage could be

intended to prove the first statement to a single person who accepts its truth on blind faith or incomplete

experience, and simultaneously to shed light on this truth. Thus, the passage can be correctly interpreted

as both an explanation and an argument.

Perhaps the greatest problem confronting the effort to distinguish explanations from arguments lies in

determining whether something is an accepted matter of fact.

Obviously what is accepted by one person may not be accepted by another. Thus, the effort often involves

determining which person or group of people the passage is directed to—the intended audience.

Sometimes the source of the passage (textbook, newspaper, technical journal, etc.) will decide the issue.

But when the passage is taken totally out of context, this may prove impossible. In those circumstances

the only possible answer may be to say that if the passage is an argument, then such-and-such is the

conclusion and such-and-such are the premises.

9. Conditional Statements: A conditional statement is an ‘‘if . . . then . . .’’ statement; for example:

If air is removed from a solid closed container, then the container will weigh less than it did.

Every conditional statement is made up of two component statements. The component statement

immediately following the ‘‘if’’ is called the antecedent, and the one following the ‘‘then’’ is called the

consequent. (Occasionally, the word ‘‘then’’ is left out, and occasionally the order of antecedent and

consequent is reversed.) In the above example the antecedent is ‘‘Air is removed from a solid closed

container,’’ and the consequent is ‘‘the container will weigh less than it did.’’ This example asserts a

causal connection between the air being removed and the container weighing less.

However, not all conditional statements express causal connections. The statement ‘‘if yellow fever is an

infectious disease, then the Dallas Cowboys are a football team’’ is just as much a conditional statement

as the one about the closed container.

Conditional statements are not arguments, because they fail to meet the criteria given earlier. In an

argument, at least one statement must claim to present evidence, and there must be a claim that this

evidence implies something. In a conditional statement, there is no claim that either the antecedent or the

consequent presents evidence. In other words, there is no assertion that either the antecedent or the

consequent is true. Rather, there is only the assertion that if the antecedent is true, then so is the

consequent. Of course, a conditional statement as a whole may present evidence because it asserts a

relationship between statements. Yet when conditional statements are taken in this sense, there is still no

argument, because there is then no separate claim that this evidence implies anything.

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Some conditional statements are similar to arguments, however, in that they express the outcome of a

reasoning process. As such, they may be said to have a certain inferential content. Consider the following:

If both Saturn and Uranus have rings, then Saturn has rings.

If iron is less dense than mercury, then it will float in mercury.

The link between the antecedent and consequent of these conditional statements resembles the inferential

link between the premises and conclusion of an argument. Yet there is a difference because the premises

of an argument are claimed to be true, whereas no such claim is made for the antecedent of a conditional

statement. Accordingly, these conditional statements are not arguments. Yet their inferential content may

be re-expressed to form arguments:

Both Saturn and Uranus have rings//Therefore, Saturn has rings.

Iron is less dense than mercury//Therefore, iron will float in mercury.

Finally, while no single conditional statement is an argument, a conditional statement may serve as either

the premise or the conclusion (or both) of an argument, as the following examples illustrate:

If cigarette companies publish warning labels, then smokers assume the risk of smoking/Cigarette companies do publish warning

labels//Therefore, smokers assume the risk of smoking.

If banks make bad loans, then they will be threatened with collapse/If banks are threatened with collapse, then the taxpayer will

come to the rescue//Therefore, if banks make bad loans, then the taxpayer will come to the rescue.

The relation between conditional statements and arguments may now be summarized as follows:

1. A single conditional statement is not an argument.

2. A conditional statement may serve as either the premise or the conclusion (or both) of an argument.

3. The inferential content of a conditional statement may be reexpressed to form an argument.

The first two rules are especially pertinent to the recognition of arguments. According to the first rule, if a

passage consists of a single conditional statement, it is not an argument. But if it consists of a conditional

statement together with some other statement, then, by the second rule, it may be an argument, depending

on such factors as the presence of indicator words and an inferential relationship between the statements.

Conditional statements are especially important in logic because they express the relationship between

necessary and sufficient conditions. A is said to be a sufficient condition for B whenever the occurrence of

A is all that is needed for the occurrence of B. For example, being a dog is a sufficient condition for being

an animal. On the other hand, B is said to be a necessary condition for A whenever A cannot occur without

the occurrence of B. Thus, being an animal is a necessary condition for being a dog. These relationships

are expressed in the following conditional statements:

If X is a dog, then X is an animal.

If X is not an animal, then X is not a dog.

The first statement says that being a dog is a sufficient condition for being an animal and the second that

being an animal is a necessary condition for being a dog. However, a little reflection reveals that these

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two statements say exactly the same thing. Thus each expresses in one way a necessary condition and in

another way a sufficient condition. The terminology of sufficient and necessary conditions will be used in

later chapters to express definitions and causal connections.

Summary: In deciding whether a passage contains an argument, one should look for three things:

(1) Indicator words such as ‘‘therefore,’’ ‘‘since,’’ ‘‘because,’’ and so on; (2) an inferential relationship

between the statements; and (3) typical kinds of nonarguments. But remember that the mere occurrence of

an indicator word does not guarantee the presence of an argument. One must check to see that the

statement identified as the conclusion is intended to be supported by one or more of the other statements.

Also keep in mind that in many arguments that lack indicator words, the conclusion is the first statement.

Furthermore it helps to mentally insert the word ‘‘therefore’’ before the various statements before

deciding that a statement should be interpreted as a conclusion. The typical kinds of non-arguments that

we have surveyed are as follows:

Warnings, pieces of advice, statements of belief, statements of opinion, loosely associated statements,

reports, expository passages, illustrations, explanations, conditional statements

Keep in mind that these kinds of non-argument are not mutually exclusive, and that, for example, one and

the same passage can sometimes be interpreted as both a report and a statement of opinion, or as both an

expository passage and an illustration. The precise kind of non-argument a passage might be is nowhere

near as important as correctly deciding whether or not it is an argument.

1.3: Deduction and Induction

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Arguments can be divided into two groups: deductive and inductive. A deductive argument is an

argument in which the premises are claimed to support the conclusion in such a way that it is impossible

for the premises to be true and the conclusion false. In such arguments the conclusion is claimed to follow

necessarily from the premises. On the other hand, an inductive argument is an argument in which the

premises are claimed to support the conclusion in such a way that it is improbable that the premises be

true and the conclusion false. In these arguments the conclusion is claimed to follow only probably from

the premises. Thus, deductive arguments are those that involve necessary reasoning, and inductive

arguments are those that involve probabilistic reasoning.

Examples: The meerkat is closely related to the suricat/ the suricat thrives on beetle larvae// Therefore, probably the meerkat

thrives on beetle larvae.

The meerkat is a member of the mongoose family/All members of the mongoose family are carnivores//Therefore, it necessarily

follows that the meerkat is a carnivore.

The first of these arguments is inductive, the second deductive.

Differentiating deductive from inductive and vice-versa: Three factors that influence our decision

about this claim are (1) the occurrence of special indicator words, (2) the actual strength of the inferential

link between premises and conclusion, and (3) the character or form of argumentation the arguer uses.

1. The distinction between inductive and deductive arguments lies in the strength of an argument’s

inferential claim. In other words, the distinction lies in how strongly the conclusion is claimed to follow

from the premises. Unfortunately, however, in most arguments the strength of this claim is not explicitly

stated, so we must use our interpretive abilities to evaluate it.

2. The occurrence of special indicator words is illustrated in the examples we just considered. The word

‘‘probably’’ in the conclusion of the first argument suggests that the argument should be taken as

inductive, and the word ‘‘necessarily’’ in the conclusion of the second suggests that the second argument

be taken as deductive. Additional inductive indicators are ‘‘improbable,’’ ‘‘plausible,’’ ‘‘implausible,’’

‘‘likely,’’ ‘‘unlikely,’’ and ‘‘reasonable to conclude.’’ Additional deductive indicators are ‘‘certainly,’’

‘‘absolutely,’’ and ‘‘definitely.’’ (Note that the phrase ‘‘it must be the case that’’ is ambiguous; ‘‘must’’

can indicate either probability or necessity).

Inductive and deductive indicator words often suggest the correct interpretation.

However, if they conflict with one of the other criteria (discussed shortly), we should probably ignore

them. Arguers often use phrases such as ‘‘it certainly follows that’’ for rhetorical purposes to add impact

to their conclusion and not to suggest that the argument be taken as deductive. Similarly, some arguers,

not knowing the distinction between inductive and deductive, will claim to ‘‘deduce’’ a conclusion when

their argument is more correctly interpreted as inductive.

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The second factor that bears upon our interpretation of an argument as inductive or deductive is the actual

strength of the inferential link between premises and conclusion.

If the conclusion actually does follow with strict necessity from the premises, the argument is clearly

deductive. In such an argument it is impossible for the premises to be true and the conclusion false. On

the other hand, if the conclusion does not follow with strict necessity but does follow probably, it is

usually best to consider the argument inductive.

Examples:

All saleswomen are extroverts/Elizabeth Taylor is a saleswoman//Therefore, Elizabeth Taylor is an extrovert.

The vast majority of saleswomen are extroverts/Elizabeth Taylor is a saleswoman//Therefore, Elizabeth Taylor is an extrovert.

In the first example, the conclusion follows with strict necessity from the premises. If we assume that all

saleswomen are extroverts and that Elizabeth Taylor is a saleswoman, then it is impossible that Elizabeth

Taylor not be an extrovert. Thus, we should interpret this argument as deductive. In the second example,

the conclusion does not follow from the premises with strict necessity, but it does follow with some

degree of probability. If we assume that the premises are true, then based on that assumption it is

improbable that the conclusion is false. Thus, it is best to interpret the second argument as inductive.

3. Occasionally, an argument contains no indicator words, and the conclusion does not follow either

necessarily or probably from the premises; in other words, it does not follow at all. This situation points

up the need for the third factor to be taken into account, which is the character or form of argumentation

the arguer uses.

a. Five examples of argumentation that are typically deductive are arguments based on mathematics,

arguments from definition, and categorical, hypothetical, and disjunctive syllogisms. Additional ones will

be addressed in later chapters.

An argument based on mathematics is an argument in which the conclusion depends on some purely

arithmetic or geometric computation or measurement. For example, a shopper might place two apples and

three oranges into a paper bag and then conclude that the bag contains five pieces of fruit. Or a surveyor

might measure a square piece of land and, after determining that it is 100 feet on each side, conclude that

it contains 10,000 square feet. Since all arguments in pure mathematics are deductive, we can usually

consider arguments that depend on mathematics to be deductive as well. A noteworthy exception,

however, is arguments that depend on statistics. As we will see shortly, such arguments are usually best

interpreted as inductive.

An argument from definition is an argument in which the conclusion is claimed to depend merely upon

the definition of some word or phrase used in the premise or conclusion. For example, someone might

argue that because Claudia is mendacious, it follows that she tells lies, or that because a certain paragraph

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is prolix, it follows that it is excessively wordy. These arguments are deductive because their conclusions

follow with necessity from the definitions of ‘‘mendacious’’ and ‘‘prolix.’’

A syllogism, in general, is an argument consisting of exactly two premises and one conclusion.

Categorical syllogisms will be treated in greater depth in Chapter 5, but for now we will say that a

categorical syllogism is a syllogism in which each statement begins with one of the words ‘‘all,’’ ‘‘no,’’

or ‘‘some.’’

Example: All lasers are optical devices//Some lasers are surgical instruments//Therefore, some optical devices are surgical

instruments.

Arguments such as these are nearly always best treated as deductive.

A hypothetical syllogism is a syllogism having a conditional statement for one or both of its premises.

Examples: If electricity flows through a conductor, then a magnetic field is produced/If a magnetic field is produced, then a

nearby compass will be deflected//Therefore, if electricity flows through a conductor, then a nearby compass will be deflected.

If quartz scratches glass, then quartz is harder than glass/Quartz scratches glass//Therefore, quartz is harder than glass.

Although certain forms of such arguments can sometimes be interpreted inductively, the deductive

interpretation is usually the most appropriate.

A disjunctive syllogism is a syllogism having a disjunctive statement (i.e., an ‘‘either . . . or . . .’’

statement) for one of its premises.

Example: Either breach of contract is a crime or it is not punishable by the state/Breach of contract is not a crime//Therefore, it

is not punishable by the state.

Like hypothetical syllogisms, such arguments are usually best taken as deductive. Hypothetical and

disjunctive syllogisms will be treated in greater depth in Chapter 6.

b. Now let us consider some typically inductive forms of argumentation. In general, inductive arguments

are such that the content of the conclusion is in some way intended to ‘‘go beyond’’ the content of the

premises. The premises of such an argument typically deal with some subject that is relatively familiar,

and the conclusion then moves beyond this to a subject that is less familiar or that little is known about.

Such an argument may take any of several forms: predictions about the future, arguments from analogy,

inductive generalizations, arguments from authority, arguments based on signs, and causal inferences, to

name just a few.

In a prediction, the premises deal with some known event in the present or past, and the conclusion

moves beyond this event to some event in the relative future. For example, someone might argue that

because certain meteorological phenomena have been observed to develop over a certain region of central

Missouri, a storm will occur there in six hours. Or again, one might argue that because certain

fluctuations occurred in the prime interest rate on Friday, the value of the dollar will decrease against

foreign currencies on Monday. Nearly everyone realizes that the future cannot be known with certainty;

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thus, whenever an argument makes a prediction about the future, one is usually justified in considering

the argument inductive.

An argument from analogy is an argument that depends on the existence of an analogy, or similarity,

between two things or states of affairs. Because of the existence of this analogy, a certain condition that

affects the better-known thing or situation is concluded to affect the similar, lesser-known thing or

situation. For example, someone might argue that because Christina’s Porsche is a great handling car, it

follows that Angela’s Porsche must also be a great handling car. The argument depends on the existence

of a similarity, or analogy, between the two cars. The certitude attending such an inference is obviously

probabilistic at best.

An inductive generalization is an argument that proceeds from the knowledge of a selected sample to

some claim about the whole group. Because the members of the sample have a certain characteristic, it is

argued that all the members of the group have that same characteristic. For example, one might argue that

because three oranges selected from a certain crate were especially tasty and juicy, all the oranges from

that crate are especially tasty and juicy. Or again, one might argue that because six out of a total of nine

members sampled from a certain labor union intend to vote for Johnson for union president, two-thirds of

the entire membership intends to vote for Johnson. These examples illustrate the use of statistics in

inductive argumentation.

An argument from authority is an argument in which the conclusion rests upon a statement made by

some presumed authority or witness. For example, a person might argue that earnings for Hewlett-

Packard Corporation will be up in the coming quarter because of a statement to that effect by an

investment counselor. Or a lawyer might argue that Mack the Knife committed the murder because an

eyewitness testified to that effect under oath. Because the investment counselor and the eyewitness could

be either mistaken or lying, such arguments are essentially probabilistic.

An argument based on signs is an argument that proceeds from the knowledge of a certain sign to

knowledge of the thing or situation that the sign symbolizes. For example, when driving on an unfamiliar

highway one might see a sign indicating that the road makes several sharp turns one mile ahead. Based on

this information, one might argue that the road does indeed make several sharp turns one mile ahead.

Because the sign might be misplaced or erroneous about the turns, the conclusion is only probable.

A causal inference underlies arguments that proceed from knowledge of a cause to knowledge of the

effect, or, conversely, from knowledge of an effect to knowledge of a cause. For example, from the

knowledge that a bottle of wine had been accidentally left in the freezer overnight, someone might

conclude that it had frozen (cause to effect). Conversely, after tasting a piece of chicken and finding it dry

and crunchy, one might conclude that it had been overcooked (effect to cause). Because specific instances

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of cause and effect can never be known with absolute certainty, one may usually interpret such arguments

as inductive.

It should be noted that the various subspecies of inductive arguments listed here are not intended to be

mutually exclusive. Overlaps can and do occur. For example, many causal inferences that proceed from

cause to effect also qualify as predictions.

The purpose of this survey is not to demarcate in precise terms the various forms of induction but rather

to provide guidelines for distinguishing induction from deduction. Keeping this in mind, we should take

care not to confuse arguments in geometry, which are always deductive, with arguments from analogy or

inductive generalizations.

For example, an argument concluding that a triangle has a certain attribute (such as a right angle)

because another triangle, with which it is congruent, also has that attribute might be mistaken for an

argument from analogy. Similarly, an argument that concludes that all triangles have a certain attribute

(such as angles totaling two right angles) because any particular triangle has that attribute might be

mistaken for an inductive generalization. Arguments such as these, however, are always deductive,

because the conclusion follows necessarily and with complete certainty from the premises.

One broad classification of arguments not listed in this survey is scientific arguments.

Arguments that occur in science can be either inductive or deductive, depending on the circumstances. In

general, arguments aimed at the discovery of a law of nature are usually considered inductive. Suppose,

for example, that we want to discover a law that governs the time required for a falling body to strike the

earth. We drop bodies of various weights from various heights and measure the time it takes them to fall.

Comparing our measurements, we notice that the time is approximately proportional to the square root of

the distance. From this we conclude that the time required for any body to fall is proportional to the

square root of the distance through which it falls. Such an argument is best interpreted as an inductive

generalization.

Another type of argument that occurs in science has to do with the application of known laws to specific

circumstances. Arguments of this sort are often considered to be deductive—but only with certain

reservations. Suppose, for example, that we want to apply Boyle’s law for ideal gases to a container of

gas in our laboratory. Boyle’s law states that the pressure exerted by a gas on the walls of its container is

inversely proportional to the volume. Applying this law, we conclude that when we reduce the volume of

our laboratory sample by half, we will double the pressure. Considered purely as a mathematical

computation, this argument is deductive. But if we acknowledge the fact that the conclusion pertains to

the future and the possibility that Boyle’s law may not work in the future, then the argument is best

considered inductive.

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A final point needs to be made about the distinction between inductive and deductive arguments. There is

a tradition extending back to the time of Aristotle which holds that inductive arguments are those that

proceed from the particular to the general, while deductive arguments are those that proceed from the

general to the particular. (A particular statement is one that makes a claim about one or more particular

members of a class, while a general statement makes a claim about all the members of a class.) It is true,

of course, that many inductive and deductive arguments do work in this way; but this fact should not be

used as a criterion for distinguishing induction from deduction. As a matter of fact, there are deductive

arguments that proceed from the general to the general, from the particular to the particular, and from the

particular to the general, as well as from the general to the particular; and there are inductive arguments

that do the same. For example, here is a deductive argument that proceeds from the particular to the

general:

Three is a prime number/Five is a prime number/Seven is a prime number//Therefore, all odd numbers between two and eight are

prime numbers.

And here is one that proceeds from the particular to the particular:

Gabriel is a wolf/Gabriel has a tail//Therefore, Gabriel’s tail is the tail of a wolf.

Here is an inductive argument that proceeds from the general to the particular:

All emeralds previously found have been green//Therefore, the next emerald to be found will be green.

The other varieties are easy to construct. Thus, the progression from particular to general, and vice versa,

cannot be used as a criterion for distinguishing induction from deduction.

In summary, to distinguish deductive arguments from inductive, we look for special indicator words, the

actual strength of the inferential link between premises and conclusion, and the character or form of

argumentation. If the conclusion follows with strict necessity from the premises, the argument is always

deductive; if not, it could be either deductive or inductive depending on the other factors. The deductive

and inductive arguments that we have surveyed in this section are as follows:

Deductive arguments: arguments based on mathematics, arguments from definition, categorical syllogisms, hypothetical

syllogisms, and disjunctive syllogisms

Inductive arguments: predictions, arguments from analogy, inductive generalizations, arguments from authority,

arguments based on signs, and causal inferences

1.4 Argument Evaluation: Validity, Truth, Soundness, Strength, Cogency

This section introduces the central ideas and terminology required to evaluate arguments.

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We have seen that every argument makes two basic claims: a claim that evidence or reasons exist and a

claim that the alleged evidence or reasons support something (or that something follows from the alleged

evidence or reasons). The first is a factual claim, the second an inferential claim. The evaluation of every

argument centers on the evaluation of these two claims. The most important of the two is the inferential

claim, because if the premises fail to support the conclusion (that is, if the reasoning is bad), an argument

is worthless. Thus we will always test the inferential claim first, and only if the premises do support the

conclusion will we test the factual claim (that is, the claim that the premises present genuine evidence, or

are true). The material that follows considers first deductive arguments and then inductive.

Deductive Arguments: The previous section defined a deductive argument as one in which the

premises are claimed to support the conclusion in such a way that it is impossible for the premises to be

true and the conclusion false. If the premises do in fact support the conclusion in this way, the argument is

said to be valid. Thus, a valid deductive argument is an argument such that it is impossible for the

premises to be true and the conclusion false. In these arguments the conclusion follows with strict

necessity from the premises.

Conversely, an invalid deductive argument is a deductive argument such that it is possible for the

premises to be true and the conclusion false. In invalid arguments the conclusion does not follow with

strict necessity from the premises, even though it is claimed to.

An immediate consequence of these definitions is that there is no middle ground between valid and

invalid. There are no arguments that are ‘‘almost’’ valid and ‘‘almost’’ invalid. If the conclusion follows

with strict necessity from the premises, the argument is valid; if not, it is invalid.

To test an argument for validity we begin by assuming that all premises are true, and then we determine if

it is possible, in light of that assumption, for the conclusion to be false.

Here is an example: All television networks are media companies/NBC is a television network//Therefore, NBC is a media

company.

In this argument both premises are actually true, so it is easy to assume that they are true. Next we

determine, in light of this assumption, if it is possible for the conclusion to be false. Clearly this is not

possible. If NBC is included in the group of television networks (second premise) and if the group of

television networks is included in the group of media companies (first premise), it necessarily follows that

NBC is included in the group of media companies (conclusion). In other words, assuming the premises

true and the conclusion false entails a strict contradiction. Thus the argument is valid.

Here is another example:

All automakers are computer manufacturers/United Airlines is an automaker//Therefore, United Airlines is a computer

manufacturer.

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In this argument, both premises are actually false, but it is easy to assume that they are true. Every

automaker could have a corporate division that manufactures computers.

Also, in addition to flying airplanes, United Airlines could make cars. Next, in light of these assumptions,

we determine if it is possible for the conclusion to be false. Again, we see that this is not possible, by the

same reasoning as the previous example.

Assuming the premises true and the conclusion false entails a contradiction. Thus, the argument is valid.

Another example:

All banks are financial institutions/Wells Fargo is a financial institution//Therefore, Wells Fargo is a bank.

As in the first example, both premises of this argument are true, so it is easy to assume they are true. Next

we determine, in light of this assumption, if it is possible for the conclusion to be false. In this case it is

possible. If banks were included in one part of the group of financial institutions and Wells Fargo were

included in another part, then Wells Fargo would not be a bank. In other words, assuming the premises

true and the conclusion false does not involve any contradiction, and so the argument is invalid.

In addition to illustrating the basic idea of validity, these examples suggest an important point about

validity and truth. In general, validity is not something that is determined by the actual truth or falsity of

the premises and conclusion. Both the NBC example and theWells Fargo example have actually true

premises and an actually true conclusion, yet one is valid and the other invalid. The United Airlines

example has actually false premises and an actually false conclusion, yet the argument is valid.

Rather, validity is something that is determined by the relationship between premises and conclusion. The

question is not whether premises and conclusion are true or false, but whether the premises support the

conclusion. In the examples of valid arguments the premises do support the conclusion, and in the invalid

case they do not. Nevertheless, there is one arrangement of truth and falsity in the premises and

conclusion that does determine the issue of validity. Any deductive argument having actually true

premises and an actually false conclusion is invalid. The reasoning behind this fact is fairly obvious. If the

premises are actually true and the conclusion is actually false, then it certainly is possible for the premises

to be true and the conclusion false. Thus, by the definition of invalidity, the argument is invalid.

The idea that any deductive argument having actually true premises and a false conclusion is invalid may

be the most important point in all of deductive logic. The entire system of deductive logic would be quite

useless if it accepted as valid any inferential process by which a person could start with truth in the

premises and arrive at falsity in the conclusion.

Table 1.1 presents examples of deductive arguments that illustrate the various combinations of truth and

falsity in the premises and conclusion.

Table 1.1 Deductive Arguments

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Valid Invalid

TruepremiseProbablytrueconclusion

All wines are beverages.Chardonnay is a wine.Therefore, chardonnay isa beverage.[sound]

All wines are beverages.Chardonnay is a beverage.Therefore, chardonnay is a wine.(unsound)

TruepremiseProbablyfalseconclusion

None existAll wines are beverages.Ginger ale is a beverage.Therefore, ginger ale is a wine.[unsound]

FalsepremiseProbablytrueconclusion

All wines are soft drinks.Ginger ale is a wine.Therefore, ginger ale is asoft drink.[unsound]

All wines are whiskeys.Chardonnay is a whiskey.Therefore, chardonnay is a wine.[unsound]

FalsepremiseProbablyfalseconclusion

All wines are whiskeys.Ginger ale is a wine.Therefore, ginger ale isa whiskey.[unsound]

All wines are whiskeys.Ginger ale is a whiskey.Therefore, ginger ale is a wine.[unsound]

In the examples having false premises, both premises are false, but it is easy to construct other examples

having only one false premise. When examining this table, note that the only combination of truth and

falsity that does not allow for both valid and invalid arguments is true premises and false conclusion. As

we have just seen, any argument having this combination is necessarily invalid.

The relationship between the validity of a deductive argument and the truth or falsity of its premises and

conclusion, as illustrated in Table 1.1, is summarized as follows:

Premises conclusion Evaluation

T T ?

T F DEFINITELY INVALID

F T ?

F F ?

A sound argument is a deductive argument that is valid and has all true premises. Thus: Valid

Deductive Argument + All True Premises = Sound Argument

Both conditions must be met for an argument to be sound, and if either is missing the argument is

unsound. Thus, an unsound argument is a deductive argument that is invalid, has one or more false

premises, or both. Because a valid argument is one such that it is impossible for the premises to be true 24

and the conclusion false, and because a sound argument does in fact have true premises, it follows that

every sound argument, by definition, will have a true conclusion as well. A sound argument, therefore, is

what is meant by a ‘‘good’’ deductive argument in the fullest sense of the term.

In connection with this definition of soundness, a single proviso is required: For an argument to be

unsound, the false premise or premises must actually be needed to support the conclusion. An argument

with a conclusion that is validly supported by true premises but with a superfluous false premise would

still be sound. Analogous remarks, incidentally, extend to induction.

Inductive Arguments: Section 1.3 defined an inductive argument as one in which the premises are

claimed to support the conclusion in such a way that it is improbable that the premises be true and the

conclusion false. If the premises do in fact support the conclusion in this way, the argument is said to be

strong. Thus, a strong inductive argument is an inductive argument such that it is improbable that the

premises be true and the conclusion false.

In such arguments, the conclusion follows probably from the premises. Conversely, a weak inductive

argument is an inductive argument such that the conclusion does not follow probably from the premises,

even though it is claimed to. The procedure for testing the strength of inductive arguments runs parallel to

the procedure for deduction. First we assume the premises are true, and then we determine whether, based

on that assumption, the conclusion is probably true.

Example: All dinosaur bones discovered to this day have been at least 50 million years old/Therefore, probably the next

dinosaur bone to be found will be at least 50 million years old.

In this argument the premise is actually true, so it is easy to assume that it is true.

Based on that assumption, the conclusion is probably true, so the argument is strong.

Here is another example: All meteorites found to this day have contained gold. Therefore, probably the next meteorite to

be found will contain gold.

The premise of this argument is actually false. Few, if any, meteorites contain any gold. But if we assume

the premise is true, then based on that assumption, the conclusion would probably be true. Thus, the

argument is strong.

The next example is an argument from analogy: When a lighted match is slowly dunked into water, the flame is

snuffed out. But gasoline is a liquid, just like water. Therefore, when a lighted match is slowly dunked into gasoline, the flame

will be snuffed out.

In this argument the premises are actually true and the conclusion is probably false.

Thus, if we assume the premises are true, then, based on that assumption, it is not probable that the

conclusion is true. Thus, the argument is weak.

Another example: During the past fifty years, inflation has consistently reduced the value of the American dollar. Therefore,

industrial productivity will probably increase in the years ahead.

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In this argument, the premise is actually true and the conclusion is probably true in the actual world, but

the probability of the conclusion is in no way based on the assumption that the premise is true. Because

there is no direct connection between inflation and increased industrial productivity, the premise is

irrelevant to the conclusion and it provides no probabilistic support for it. The conclusion is probably true

independently of the premise. As a result, the argument is weak.

This last example illustrates an important distinction between strong inductive arguments and valid

deductive arguments. As we will see in later chapters, if the conclusion of a deductive argument is

necessarily true independently of the premises, the argument will still be considered valid. But if the

conclusion of an inductive argument is probably true independently of the premises, the argument will be

weak.

These four examples show that in general the strength or weakness of an inductive argument results not

from the actual truth or falsity of the premises and conclusion, but from the probabilistic support the

premises give to the conclusion. The dinosaur argument has a true premise and probably true conclusion,

and the meteorite argument has a false premise and a probably false conclusion; yet, both are strong

because the premise of each provides probabilistic support for the conclusion. The industrial productivity

argument has a true premise and a probably true conclusion, but the argument is weak because the

premise provides no probabilistic support for the conclusion.

Analogously to the evaluation of deductive arguments, the only arrangement of truth and falsity that

establishes anything is true premises and probably false conclusion (as in the lighted match argument).

Any inductive argument having true premises and a probably false conclusion is weak.

Table 1.2 presents the various possibilities of truth and falsity in the premises and conclusion of inductive

arguments. Note that the only arrangement of truth and falsity that is missing for strong arguments is true

premises and probably false conclusion.

The relationship between the strength of an inductive argument and the truth or falsity of its premises and

conclusion, as illustrated in Table 1.2, is summarized as follows:

Premises conclusion Evaluation

T Probably T ?

T Probably F weak

F Probably T ?

F Probably F ?

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Table 1.2 Inductive Arguments

Strong weak

TruepremiseProbablytrueconclusion

All previous American presidents were men.Therefore, probably the nextAmerican president will be a man.[cogent]

A few American presidents were Federalists.Therefore, probably the nextAmerican president will be a man.[uncogent]

TruepremiseProbablyfalseconclusion

None existA few American presidents were Federalists.Therefore, probably the nextAmerican president will be aFederalist.[uncogent]

FalsepremiseProbablytrueconclusion

All previous American presidents were television debaters.Therefore, probably the nextAmerican president will be a television debater.[uncogent]

A few American presidents wereLibertarians.Therefore, probably the nextAmerican president will be a television debater.[uncogent]

FalsepremiseProbablyfalseconclusion

All previous American presidents were women.Therefore, probably the nextAmerican president will be a woman.[uncogent]

A few American presidents wereLibertarians.Therefore, probably the nextAmerican president will be aLibertarian.[uncogent]

Unlike the validity and invalidity of deductive arguments, the strength and weakness of inductive

arguments admit of degrees. To be considered strong, an inductive argument must have a conclusion that

is more probable than improbable. In other words, the likelihood that the conclusion is true must be more

than 50 percent, and as the probability increases, the argument becomes stronger. For this purpose,

consider the following pair of arguments:

This barrel contains 100 apples/Three apples selected at random were found to be ripe//Therefore, probably all 100 apples are

ripe/This barrel contains 100 apples/Eighty apples selected at random were found to be ripe//Therefore, probably all 100 apples

are ripe.

The first argument is weak and the second is strong. However, the first is not absolutely weak nor the

second absolutely strong. Both arguments would be strengthened or weakened by the random selection of

a larger or smaller sample. For example, if the size of the sample in the second argument were reduced to

70 apples, the argument would be weakened. The incorporation of additional premises into an inductive

argument will also generally tend to strengthen or weaken it. For example, if the premise ‘‘One unripe

apple that had been found earlier was removed’’ were added to either argument, the argument would be

weakened.

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A cogent argument is an inductive argument that is strong and has all true premises; if either condition

is missing, the argument is uncogent. Thus, an uncogent argument is an inductive argument that is weak,

has one or more false premises, or both.

A cogent argument is the inductive analogue of a sound deductive argument and is what is meant by a

‘‘good’’ inductive argument without qualification. Because the conclusion of a cogent argument is

genuinely supported by true premises, it follows that the conclusion of every cogent argument is probably

true.

Cogent argument = Strong argument + All true premises. If either condition is missing the argument is

uncogent.

There is a difference, however, between sound and cogent arguments in regard to the true-premise

requirement. In a sound argument it is only necessary that the premises be true and nothing more. Given

such premises and good reasoning, a true conclusion is guaranteed. In a cogent argument, on the other

hand, the premises must not only be true, they must also not ignore some important piece of evidence that

outweighs the given evidence and entails a quite different conclusion. As an illustration of this point,

consider the following argument:

Swimming in the Caribbean is usually lots of fun. Today the water is warm, the surf is gentle, and on this beach there are no

dangerous currents. Therefore, it would be fun to go swimming here now.

If the premises reflect all the important factors, then the argument is cogent. But if they ignore the fact

that several large dorsal fins are cutting through the water, then obviously the argument is not cogent.

Thus, for cogency the premises must not only be true but also not overlook some important factor that

outweighs the given evidence and requires a different conclusion.

In summary, for both deductive and inductive arguments, two separate questions need to be answered: (1)

Do the premises support the conclusion? (2) Are all the premises true?

To answer the first question we begin by assuming the premises to be true. Then, for deductive arguments

we determine whether, in light of this assumption, it necessarily follows that the conclusion is true. If it

does, the argument is valid; if not, it is invalid. For inductive arguments we determine whether it probably

follows that the conclusion is true. If it does, the argument is strong; if not, it is weak. For inductive

arguments we keep in mind the requirements that the premises actually support the conclusion and that

they not ignore important evidence. Finally, if the argument is either valid or strong, we turn to the second

question and determine whether the premises are actually true. If all the premises are true, the argument is

sound (in the case of deduction) or cogent (in the case of induction). All invalid deductive arguments are

unsound, and all weak inductive arguments are uncogent.

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The various alternatives open to statements and arguments may be diagrammed as follows. Note that in

logic one never speaks of an argument as being ‘‘true’’ or ‘‘false,’’ and one never speaks of a statement

as being ‘‘valid,’’ ‘‘invalid,’’ ‘‘strong,’’ or ‘‘weak.’’

Summary of Module 1

Logic is the study of the evaluation of arguments, which are lists of statements

consisting of one or more premises and one conclusion. Premises can be

distinguished from conclusion by the occurrence of indicator words (‘‘hence,’’

‘‘therefore,’’ ‘‘since,’’ and so on) or an inferential relation among the statements.

Because not all groups of statements are arguments, it is important to be able to

distinguish arguments from nonarguments. This is done by attending to indicator

words, the presence of an inferential relation among the statements, and typical

kinds of nonarguments. Typical nonExtended arguments include warnings, loosely

associated statements, reports, expository passages, illustrations, conditional

statements, and explanations.

Arguments are customarily divided into deductive and inductive. Deductive

arguments are those in which the conclusion is claimed to follow necessarily from

the premises, while inductive arguments are those in which the conclusion is

claimed to follow only probably from the premises. The two can be distinguished by

attending to special indicator words (‘‘it necessarily follows that,’’ ‘‘it probably

follows that,’’ and so on), the actual strength of the inferential relation, and typical

forms or styles of deductive and inductive argumentation. Typical deductive

arguments include arguments based on mathematics, arguments from definition,

and categorical, hypothetical, and disjunctive syllogisms. Typical inductive

arguments include predictions, arguments from analogy, generalizations,

arguments from authority, arguments based on signs, and causal inferences.

The evaluation of arguments involves two steps: evaluating the link between

premises and conclusion, and evaluating the truth of the premises. Deductive

arguments in which the conclusion actually follows from the premises are said to be

valid, and those that also have true premise are said to be sound. Inductive

arguments in which the conclusion actually follows from the premises are said to be

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strong, and those that also have true premises are said to be cogent. The terms

‘‘true’’ and ‘‘false’’ apply not to arguments, but to statements. The truth and falsity

of premises and conclusion is only indirectly related to validity, but any deductive

argument having true premises and false conclusion is invalid.

Exercises: Pick Hurley’s Text, preferably editions from 5 onwards, and answer each of the following

sections separately!

Section 1.1: Answer III and IV as correctly and completely as possible (each separately!)

Section 1.2: Attempt I, II, III and VI.

: Answer IV and V as correctly and completely as possible (each separately!)

Section 1.3: Attempt I (unanswered questions)

: Answer II and III as completely and correctly as possible (each separately!)

Section 1.4: Attempt all unanswered question under I, II, and III.

: Answer IV and V as completely and correctly as possible (each separately!)

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